Partitioning finite d-dimensional integer grids with applications

Abstract

In this chapter we will consider partitions of finite d-dimensional integer grids, that is, sets of the form {0, 1, …, m−1}d, by lines in two-dimensional space or by hyperplanes and hypersurfaces in an arbitrary dimension. Different aspects of the problem depending on m, d, and the type of hypersurfaces used have been widely studied in different areas of computer science and mathematics. In this chapter we will focus on problems arising in the areas of digital image processing (analysis) and neural networks. For brevity, related problems arising in other areas of computing (e.g., multivalued logic) and in pure mathematics areas (e.g., group theory) will not be analyzed.